×

On some spaces isomorphic to the space of absolutely \(q\)-summable double sequences. (English) Zbl 1441.46001

Summary: Let \(0 < q < \infty\). In this study, we introduce the spaces \(\mathcal{BV}_q\) and \(\mathcal{LS}_q\) of \(q\)-bounded variation double sequences and \(q\)-summable double series as the domain of four-dimensional backward difference matrix \(\Delta\) and summation matrix \(S\) in the space \(\mathcal{L}_q\) of absolutely \(q\)-summable double sequences, respectively. Also, we determine their \(\alpha\)- and \(\beta\)-duals and give the characterizations of some classes of four-dimensional matrix transformations in the case \(0 < q \leq 1\).

MSC:

46A45 Sequence spaces (including Köthe sequence spaces)
40C05 Matrix methods for summability
40B05 Multiple sequences and series
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] C. R. Adams,On non-factorable transformations of double sequences, Proc. Natl. Acad. Sci. USA.,19(5)(1933), 564-567. · JFM 59.0247.01
[2] B. Altay and F. Ba¸sar,Some new spaces of double sequences, J. Math. Anal. Appl., 309(2005), 70-90. · Zbl 1093.46004
[3] B. Altay and F. Ba¸sar,The fine spectrum and the matrix domain of the difference operator∆on the sequence space‘p,(0< p <1), Commun. Math. Anal.,2(2)(2007), 1-11. · Zbl 1173.47021
[4] F. Ba¸sar and B. Altay,Matrix mappings on the spacebs(p)and itsα-,β-, andγ-duals, Aligarh Bull. Math.,21(2002), 79-91.
[5] F. Ba¸sar and B. Altay,On the space of sequences ofp-bounded variation and related matrix mappings, Ukrainian Math. J.,55(2003), 136-147. · Zbl 1040.46022
[6] F. Ba¸sar, B. Altay and M. Mursaleen,Some generalizations of the spacebvpofpbounded variation sequences, Nonlinear Anal.,68(2)(2008), 273-287. · Zbl 1132.46002
[7] F. Ba¸sar and H. C¸ apan,On the paranormed spaces of regularly convergent double sequences, Results Math.,72(1-2)(2017), 893-906. · Zbl 1391.46007
[8] F. Ba¸sar and H. C¸ apan,On the paranormed spaceMu(t)of double sequences, Bol. Soc. Parana. Mat.,37(3)(2019), 99-111. · Zbl 1424.46008
[9] F. Ba¸sar and Y. Sever,The spaceLqof double sequences, Math. J. Okayama Univ., 51(2009), 149-157. · Zbl 1168.46300
[10] H. C¸ apan and F. Ba¸sar,Some paranormed difference spaces of double sequences, Indian J. Math.,58(3)(2016), 405-427. · Zbl 1376.46009
[11] H. C¸ apan and F. Ba¸sar,On the paranormed spaceL(t)of double sequences, Filomat, 32(3)(2018), 1043-1053. · Zbl 1499.40055
[12] B. Choudhary and S.K. Mishra,On K¨othe-Toeplitz duals of certain sequence spaces and their matrix transformations, Indian J. Pure Appl. Math.,24(5)(1993), 291-301. · Zbl 0805.46008
[13] R. C. Cooke,Infinite matrices and sequence spaces, Macmillan and Co. Limited, London, 1950. · Zbl 0040.02501
[14] S. Demiriz and O. Duyar,Domain of difference matrix of order one in some spaces of double sequences, Gulf J. Math.,3(3)(2015), 85-100. · Zbl 1389.46004
[15] M. Kiri¸s¸ci and F. Ba¸sar,Some new sequence spaces derived by the domain of generalized difference matrix, Comput. Math. Appl.,60(5)(2010), 1299-1309. · Zbl 1201.40001
[16] A. Pringsheim,Zur Theorie der zweifach unendlichen Zahlenfolgen, Math. Ann., 53(1900), 289-321. · JFM 31.0249.01
[17] M. Ye¸silkayagil and F. Ba¸sar,Domain of Riesz mean in the spaceL∗s, Filomat, 31(4)(2017), 925-940. · Zbl 1488.46015
[18] M. Zeltser,
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.